# Definition:Bounded Above Mapping/Real-Valued

This page is about Bounded Above in the context of Real-Valued Function. For other uses, see Bounded Above.

## Definition

Let $f: S \to \R$ be a real-valued function.

$f$ is bounded above on $S$ by the upper bound $H$ if and only if:

$\forall x \in S: \map f x \le H$

That is, if and only if the set $\set {\map f x: x \in S}$ is bounded above in $\R$ by $H$.

## Unbounded Above

Let $f: S \to \R$ be a real-valued function.

Then $f$ is unbounded above on $S$ if and only if it is not bounded above on $S$:

$\neg \exists H \in \R: \forall x \in S: \map f x \le H$